Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
G001 — Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary).
Definition and scope
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all endpoints in range and same-direction arcs unique;actual loops=[];main loop-free directed-set convention qualified
Conditions: none
Check outer from/to order without sorting endpoints inside each arc.(u,v) and(v,u)are distinct and may coexist.The main convention forbids loops;the boundary temporarily accepts them to audit that convention,not to deny other digraph models.
ORIGINAL consecutive positions=[0, 1];ORIGINAL directed step=[0, 1]
After
EXACT oriented arc(0,1) present=Yes;reverse arc is a DIFFERENT lookup
Conditions: none
Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement.
ORIGINAL consecutive positions=[1, 2];ORIGINAL directed step=[1, 2]
After
EXACT oriented arc(1,2) present=Yes;reverse arc is a DIFFERENT lookup
Conditions: none
Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement.
ORIGINAL consecutive positions=[2, 3];ORIGINAL directed step=[2, 3]
After
EXACT oriented arc(2,3) present=Yes;reverse arc is a DIFFERENT lookup
Conditions: none
Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement.
ORIGINAL consecutive positions=[3, 4];ORIGINAL directed step=[3, 4]
After
EXACT oriented arc(3,4) present=Yes;reverse arc is a DIFFERENT lookup
Conditions: none
Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement.
ORIGINAL consecutive positions=[4, 5];ORIGINAL directed step=[4, 0]
After
EXACT oriented arc(4,0) present=Yes;reverse arc is a DIFFERENT lookup
Conditions: none
Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement.
ALL oriented steps present=Yes;empty/singleton list has no step to refute;NOT an extra nonempty-walk,simple-path or trail certificate
Conditions: none
The flag is only the conjunction of successive-step checks.Empty/singleton lists are vacuous and do not certify a separately required nonempty-walk convention.Neither full-edge coverage nor no-revisit conditions are imposed.
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.